A particle model analysing the behavioural rules underlying the collective flight of a bee swarm towards the new nest

ABSTRACT The swarming of a bee colony is guided by a small group of scout individuals, which are informed of the target destination (the new nest). However, little is known on the underlying mechanisms, i.e. on how the information is passed within the population. In this respect, we here present a discrete mathematical model to investigate these aspects. In particular, each bee, represented by a material point, is assigned its status within the colony and set to move according to individual strategies and social interactions. More specifically, we propose alternative assumptions on the flight synchronization mechanism of uninformed individuals and on the characteristic dynamics of the scout insects. Numerical realizations then point out the combinations of behavioural hypotheses resulting in collective productive movement. An analysis of the role of the scout bee percentage and of the phenomenology of the swarm in domains with structural elements is finally performed.


Introduction
The wish of describing emerging collective dynamics of populations of interacting individuals, such as birds, fishes, insects, and certain mammals, from individual behaviours has increased in the last decades the multidisciplinary interest of various research communities, e.g. biologists, ecologists, sociologists, physicists, and applied mathematicians.
For instance, the coordinated behaviour of bee swarms represents an interesting problem to be studied. Such insect populations, which are typically composed by the old queen and by 10,000-30,000 worker individuals, in fact, undergo a synchronized flight with the specific purpose of reaching a new nest site [35]. All colonies are subject to this natural phenomenon, and every year beekeepers have to deal with it in late spring and early summer. In this period, as the weather warms up and flowers begin to bloom, the colony is in fact at the peak of its capacity and ready to produce a new hive.
In this respect, each bee swarm has to face two challenges: it first needs to identify a suitable new location using a process of community site selection and then it has to move towards the chosen destination. Entering in more details, when the migrating bees leave the original hive they first temporarily settle on a tree branch a few metres away from the old nest. There, they cluster around the queen, and a small fraction of bees (called scouts) starts exploring the surrounding area. These individuals inspect possible locations for the new nest to assess their quality, in terms of volume, aspect, size, and height of entrance, and presence of structures left by other bee colonies [35,39]. Then, they return to the rest of the population and perform a waggle dance to broadcast information on the characteristics of the explored sites. Nest proposals coming from the scout bees may be different but, after some hours (sometimes days), an agreement is finally found.
The whole swarm finally takes off and compactly flies towards the chosen destination, following the guidance of the informed/scout individuals. In this respect, various assumptions have been proposed to account for the migration mechanisms underlying this leader-follower system. First, it was hypothesized that the scout bees can pilot the cloud of insects to the new home by producing the Nasonov pheromone. However, a subsequent experimental study revealed that such a chemical substance is not really involved in the flight guidance process, while it is crucial to help the uninformed/follower individuals to find the entrance of the new nest [1].
A different family of possible explanations instead involves selected visual signals in the transmission of information within the swarm. Specifically, Lindauer in 1955 proposed that the scout insects can transmit the direction of movement by flying through the swarm [31].
In particular, such informed bees are observed to streak at high speed from the back of the swarm to its front. However, once they have reached the front of the cloud, their behaviour is still unknown. In this respect, in [33,35], two possible dynamics are suggested: (i) they may slowly fly back towards the rear edge of the swarm or (ii) they may stop and wait to be passed by the rest of the groupmates. In both cases, the scout bees then start again to streak towards the leading edge of the insect cloud and the process is iterated. The uninformed bees are in turn observed to align their flight towards the position of the new nest. How the follower individuals acquire the information of the correct migration direction under the scout guidance is debated as well. More specifically, each different hypothesis on their flight synchronization mechanism involves the alignment with a distinct set of groupmates, composed, e.g. by faster individuals, or by closer individuals, or only by informed individuals.

Objective and structure of the work
Our aim is to investigate the individual behaviour that is at the basis of bee swarming towards a new nest. In this respect, we will reproduce the dynamics of a bee population with a first-order microscopic discrete model that focuses on the flight phenomenology of each component insect.
In more details, our model will be based on the so-called first sociological principles of swarming which state that the individuals forming a flying group are subject at least to repulsive, attractive, and alignment stimuli. These assumptions have been previously implemented in a number of approaches dealing with the collective dynamics of swarms, as commented in the conclusive section of this paper and reviewed by Carrillo and coworkers in [9]. We have also employed such phenomenological rules in a previous work [2], that has .d how a single leader bee is able to transmit the direction of movement to the rest of the population. In particular, we have therein tested alternative alignment hypotheses, i.e. topological vs. metric. The former implies that each insect synchronizes its flight to a given number of groupmates regardless of their effective distance; the latter implies that each bee synchronizes its flight to all groupmates falling within a certain region. The results presented in that work have then demonstrated a directionally efficient swarming can be obtained with a sufficiently large (i.e. with radius equal to the overall extension of the insect cloud) alignment region in the case of metric alignment or with a high enough (i.e. ≥ 13) number of individuals taken into account by each bee in the case of topological alignment. In spite of such interesting results, some aspects characterizing the collective dynamics of bee clouds still remained unexplored, becoming indeed the main focus of this article.
In this respect, we will here first introduce anisotropy in insect behaviour by the definition of a more realistic visual region, i.e. not completely round. Then, we will include the experimentally observed bee differentiated behaviour within the population: we will in fact, distinguish between scout individuals and follower insects and define for each subgroup a specific phenomenology. In particular, we will test different combinations of hypotheses relative (i) to the characteristic movement of the scout/informed bees (which will be indeed described in more details with respect to our previous work, where a single leader individual was set to constantly fly in a given direction) and (ii) to the alignment mechanisms of the follower/uninformed individuals. In more details, each uninformed bee will be assumed to synchronize its movement to a given set of groupmates not only upon considerations on their mutual distance (as in our previous work) but also taking into account their status and actual velocity.
Once the most reasonable assumptions resulting in a correct collective swarming of the insect population is found out, our second objective will be then to demonstrate that the obtained behavioural rules are sufficient to reproduce realistic migratory dynamics in complex real-world scenarios, involving domains with structural elements and obstacles.
The rest of this paper is organized as follows. In Section 2, we will present the main model components. More specifically, we will first explain the characteristic representation of the virtual bees and their possible status/role within the population; then, we will write the equations of motion and introduce the relative velocity contributions. In particular, we will clarify the biological and experimental considerations each term is based on. In Section 3, selected series of numerical realizations will analyse swarm dynamics under different combinations of the proposed behavioural hypotheses of the insect colony. A study on the influence of the fraction of scout individuals on the swarming process will be performed as well. After presenting in Section 3.2 the phenomenology of the bee cloud in more realistic situations, we will review in Section 4 the results obtained in the article. In the same conclusive part of the work, we will compare our approach with similar discrete models dealing with bee dynamics presented in the literature and propose some possible improvements and developments of the work. Finally, Appendix will provide a detailed description of the parameter estimate employed in our theoretical framework.

Mathematical model
Bee characteristics and status transitions. A population of bees is modelled in the twodimensional bounded domain ⊂ R 2 , whose boundary is defined by ∂ . We are indeed considering a planar section, parallel to the ground, of a typical bee swarm, see Figure 1(a). We further assume that the new nest, i.e. the target destination of the insects, is constituted The virtual population of bees is modelled in a two-dimensional domain ⊂ R 2 , i.e. we are taking into account a planar section of a typical swarm. The aim of the insect population is to reach a new nest, which is constituted by a subregion of the domain boundary, hereafter denoted by ∂ nest (⊂ ∂ ). The domain may represent a large open-space or it may account for environmental elements, such as trees or buildings. (b) The bees within the swarm can have the following roles: U ('uninformed'), S ('streaker'), or P ('passive leader'). An uninformed bee does not change its status, i.e. it is not able to become a scout. Status transitions instead occur within the set of informed insects. More specifically, they are set to have a streaker role while flying towards the nest: in this respect, once reached the front edge of the cloud, they are assigned a passive leader role and turn the direction of flight (in the case of hypothesis L1) or stop waiting for the passage of the rest of the insect cloud (in the case of hypothesis L2). Eventually, when a passive leader finds itself at the trailing edge of the swarm, it acquires again a streaker status and move towards the target destination as well.
by a subregion of the domain boundary, hereafter denoted by ∂ nest (⊂ ∂ ), refer again to Figure 1(a). Each insect i = 1, . . . , N, being N the total number of individuals, is an autonomous discrete agent, represented by a material point with concentrated mass. In particular, the generic ith bee is uniquely defined by the following set of variables: where S 2 1 is the unit circle. The vectors x i (t), v i (t), and g i (t) denote individual position, velocity, and gazing direction, respectively, whereas s i (t) is a status variable which defines the role that the insect of interest has within the swarm. In this respect, according to the biological considerations presented in the previous section, for each bee i, s i falls in the set More specifically, scout individuals are set to have a streaker role when flying in the direction of the nest. Otherwise, they are defined as passive leaders. In this respect, we now introduce possible insect status transitions, which are summarized in panel (b) of Figure 1.
Of course, an uninformed bee can not become informed, so it can not change its status and will just follows the rest of the swarm. Status switches instead occur within the set of scout insects. In particular, let us first define for any point of the domain x ∈ the signed distance function l nest (x) : → R + ∪ {0}, which is evaluated by solving the two-dimensional eikonal equation: with boundary conditions where n is the unit vector locally normal to the domain boundary. It is useful to underline that, in other words, l nest (x) measures the length of the minimal path between any x ∈ and any point belonging to ∂ nest . Coherently with the previous experimental considerations, a streaker bee, say i, becomes a passive leader when it finds itself at the extreme frontal edge of the population, i.e. if, in mathematical terms, being vis i the visual region of individual i (see below). On the opposite, a passive leader, say again i, switches back to a streaker status if it finds itself at the trailing edge of the group of insects, i.e. if As most animal species, bees typically move and behave influenced by visual signals, captured by their large visual field, that covers almost the entire surrounding space. For the ith insect (regardless of its status), we indeed denote by the unit vector g i (t) ∈ S 2 1 (being S 2 1 the unit 2-D ball) its actual gazing direction and by vis its visual region, being d vis and θ vis the visual extension and the half visual angle, respectively (see Figure 2).For sake of simplicity, we hereafter assume that the gaze of each bee is constantly aligned to its velocity, i.e.
However, individual gazing direction may also evolve slightly independently from the direction of flight: therefore, a proper evolution equation for g i may be included as well, as done for instance in [13]. Bee dynamics. The collective movement of the swarm is described by a set of firstorder ordinary differential equations (ODEs), which derives from a generic second-order model under the assumption of overdamped force-velocity response, which is a consistent hypothesis for living entities (e.g. cells, animals, humans, see the comments in [16,21,34]). In this respect, the ith bee, with i = 1, . . . , N, moves according to where a control on its actual speed is done to avoid unrealistically high values, that may result also from plausible rules of movement. In this respect, v max denotes the maximal admissible bee speed. It is a round section determined by the visual depth d vis and the half visual angle θ vis , which symmetrically extends from the gazing direction of bee i, defined by the unit vector g i (t) (which, for the sake of simplicity, will be constantly aligned to the velocity v i (t)). The inclusion of an anisotropic visual field implies that each bee is not able to see and therefore to interact with the entire set of their groupmates (see, for instance, the individual k and h). For representative purposes, hereafter the virtual bees will be indicated by rigid disks centred at their actual position.
The insects then behave according to their role within the swarm. Coherently, we now define three different dynamics, each relative to an individual status. All of them are the sum of a given set of contributions, which can be in common for the entire population or characteristic of a single subgroup (and underlined in the following equations). In particular, for the generic ith bee, we have We now comment each term in Equation (8), starting from those active for all individuals.
The repulsive velocity component v avoid models the tendency of all bees of staying sufficiently far away from their neighbours, typically in order to avoid physical collisions and to maintain a minimal comfort space within the swarm. The term v group instead implements the desire of each individual to keep a connection with the groupmates, i.e. to remain close enough to the rest of the population. For the generic ith insect, regardless of its status, both behaviours are described by proper kernels H avoid ij , H group ij : R 2 × R 2 −→ R 2 , which define its pairwise interaction instances with the generic jth individual belonging to one of the following interaction sets: where r ij (t) := x j (t) − x i (t). In (9), d avoid is a measure of the comfort space that each insect tries to preserve, whereas d group is assumed to be equal to d vis , i.e. the bees tend to keep a connection with the groupmates within their visual region, see Figure 3(a). Further, we , are in fact given by the insects that i sees and whose distance falls in the ranges (0, d avoid ] and (d avoid , d group ], respectively. Finally, alternative assumptions are set for the flight synchronization mechanism of uninformed bees. However, in all cases, the insects taken into account by the ith follower individual have to locate within a distance of d align ∈ (d avoid , d group ). (bc) Plots of the pairwise attractive/repulsive interaction kernels h avoid and h group defined in Equations (11) and (12), respectively. assume that the above-introduced kernels do not depend on the specific couple of bees, i.e. H avoid ij = H avoid and H group ij = H group for any pair (i, j), that the resulting velocity contributions have an effect on the direction ideally connecting the interacting insects and finally that they depend on individual relative distance. In this respect, we can write: being again r ij (t) := x j (t) − x i (t). To implement the desired phenomenology, the function h avoid (respectively, h group ) has to be non positive (respectively, non negative) in its entire domain, i.e. h avoid : R + −→ R − ∪ {0} (respectively, h group : R + −→ R + ∪ {0}). In principle, there are many possible forms for such interaction laws. In particular, the avoidance bee behaviour is hereafter described by a classical Newtonian-type short-range hyperbolic kernel which has been employed in other particle models. For instance, it has been used by Diwold and coworkers to reproduce the collective flight of red dwarf honeybees (cf. [20], Equation (1)) and by Chen and Kolokolnikov to study predator-swarm interactions (cf. [10], Equation (1.1) and the references below). Repulsive kernels with similar trends (i.e. which go to infinity as |r ij | α , with α < 0, when |r ij | → 0, being |r ij | the distance between two interacting agents) have been implemented also in the case of discrete approaches for zebrafish embryogenesis [19] and endothelial patterning on polymers [29]. In (11), the positive parameter f avoid ∈ (0, +∞), with units m 2 /s, defines the slope of the hyperbolic function h avoid . On the other hand, long-range attraction between bees is assumed to have the following parabolic form: We do not use linear Hooke-like attraction laws, such as those introduced in some of the previously cited works dealing with swarming, e.g. [10,20], since we hypothesize that the attractive stimulus is negligible when two bees are very close and, after a maximum, it decreases again to zero at d group , which is taken to be the margin of the visual field. Finally, according to us, it is plausible that pairs of insects falling substantially apart one from each other do not have a significant mutual influence. The exact form of h group is taken such that its maximum is given by the positive coefficient f group ∈ (0, +∞), which has units m/s, and located in the middle of the interval (d avoid , d group ). Analogous attraction functions has been used in the case of other particle models relative to bee and cell dynamics, see [2,[11][12][13] and references therein. It is also useful to underline that, according to the above-introduced kernels h avoid and h group , which are plotted in Figure 3(b), and to the corresponding interaction sets N avoid and N group , two individuals do not interact (i) when they do not see each other and (ii) when they are exactly at the comfort distance d avoid .
It is finally useful to underline that the pair of coefficients f avoid and f group have a clear mathematical meaning, as commented above, but not a direct and measurable experimental counterpart. In this respect, their estimate has required a detailed parameter study, as discussed in the Appendix. We then include a velocity term that implements the intention of bees to remain sufficiently distant from the domain boundary (which may represent, e.g. architectural structures or natural obstacles). In accordance to the case of pedestrians [13], such a migratory contribution enters the picture when the ith individual (regardless of its status) is close enough, i.e. at a maximal distance hereafter defined with the coefficient d boundary , to a boundary: where is the unit vector directed from the nearest point of the domain boundary to the actual position of the insect i. l boundary (x i (t)) : → R + ∪ 0 is in fact the distance function resulting from the following eikonal equation and relative boundary conditions: We here remark that, despite the term defined in (13), proper boundary conditions are needed. More specifically, we hereafter assume that when a bee touches a part of the domain not occupied by the nest, it stops, whereas it is taken out from the simulation when reaches a point of the target nest. For each insect i, a fluctuation velocity term is added as well. It is given by a vector v rand i , whose modulus and direction are, at any time t, random variables which uniformly fall within the ranges of values [0, v mean /10] and [0, 360 • ), respectively (see below for the meaning of v mean ).
We now turn to describe the velocity components characteristic of the different bee subgroups. First, v align i is an alignment term typical of the following individuals which, being uninformed of the position of the new nest, are only able to synchronize their flight with selected sets of mates. In this respect, for the ith follower insect we define v align i where v mean denotes the characteristic speed of uninformed bees and is the mean of the velocities of the groupmates falling within the alignment set N align i (t), denoting by # its cardinality. In this respect, we propose four alternative definitions of N align i , in accordance with the different experimental hypotheses: HP A1 the ith uninformed bee synchronizes its flight with the follower and the streaker individuals that are sufficiently fast and close to its position. This results in ; HP A2 the ith uninformed bee aligns to the faster groupmates, regardless of their status, provided that they are close enough, i.e.
HP A3 the ith uninformed bee synchronizes its flight to all insects falling within a given neighbourhood, regardless of their status and speed. In mathematical terms: HP A4 the ith uninformed bee synchronizes its flight with the follower and the streaker individuals that fall within a given regions, regardless of their velocity: In all cases, d align defines the extension of the alignment region. As discussed in the following, d align ∈ (d avoid , d group ), i.e. the flight synchronization set of an individual intersects those relative to its pairwise interactions with the groupmates. This assumption implies that each uninformed bee can simultaneously align to and avoid or be attracted by another individual, see Figure 3(a). Further, we remark that, in the case of hypotheses A1 and A4, the passive leaders are not taken into account by the follower bees, as assumed in the biological literature [33,35]. However, it is useful to underline that the explanation of such a phenomenon is far to be understood. For instance, it is hypothesized that the passive leaders make themselves invisible to their groupmates by flying either close to the ground or backlight with respect to the sun or hidden in the middle the rest of the cloud (see again [33,35]). In this respect, given the absence of detailed experimental evidence and in order to avoid further model overcomplications, we here opt to focus only on the typology of the flight of the informed insects (i.e. 'back-and-forth' vs. 'go-and-stop') and not on the zones of the swarm where such characteristic movements are performed. This issue would require further empirical investigations and, from a mathematical point of view, the introduction of three-dimensional settings. However, planar domains to describe bee swarming, including the dynamics of the leader individuals, are consistently employed across the theoretical literature (refer, for instance, to [9,20]).
In Equation (8) 2 , v streak i describes the characteristic motion of the scout bees with a streaker status, i.e. when they fly at high speed in the direction of the nest thereby behaving as guidance leaders for the rest of the swarm. In particular, for the ith streaker insect (i.e. i : s i (t) = S), we set: where ∇l nest and v max have been introduced in (3) and (7), respectively. Equation (22) implies that each streaker individual performs a flight that, at each time instant t, is aligned to the direction minimizing the distance between its position and the target nest, i.e. it moves along the optimal trajectory. In this respect, we here remark that the use of eikonal equations is usually employed in methods related to the computation of optimal paths. More specifically, Equation (22) has the advantage that it can be used regardless the complexity of the domain, with straightforward extension to the case of non-planar geometries. Further comments on this aspect can be found in [13], where different approaches for evaluating individual minimal trajectories to a given target are discussed as well.
We then propose two alternative hypothesis for the characteristic behaviour of scout bees when they take a passive leader role: HP L1 on one hand, they are assumed to slowly fly back towards the rear edge of the swarm, in order to slightly affect the movement of the rest of the groupmates. In wherek is such that i.e.k is the uninformed insect farthest from the target nest; HP L2 on the other hand, we hypothesize that the passive leaders stop and wait for the passage of the rest of the population. For the ith passive leader, we indeed set v passive i The entire model parameter setting used in the following simulations is summarized in Table 1, while in the Appendix we will detail how they have been estimated.

Numerical results
The numerical results proposed in this section will be divided in two parts. In the first Section 3.1, we will test different combinations of the alternative assumptions relative to the alignment mechanism of the uninformed bees and to the behaviour of the passive leaders. By considering swarm dynamics within a simple rectangular domain, we will look at the rules of motion that result in a realistic migration of the insect cloud, which has to fly compactly and productively towards the nest. The effect of variations in the percentage of informed bees will be analysed as well. Section 3.2 will be instead devoted to reproduce the collective phenomenology of the bee population in more complex environments, which involve domains with different obstacles.

Swarming in a large open-space domain
In this first set of simulations, we use a rectangular [0, 200] × [0, 200] m 2 domain , where the target destination is constituted by the boundary segment y ∈ [95, 105] on the right side of the domain, see Figure 4. The measure of ∂ nest is larger than the dimension of a real nest, since we here intend to describe the behaviour of the swarm while approaching the new home and not the subsequent entrance mechanisms, that are driven by other processes (e.g. pheromone cues). The insect population is formed by N = 500 individuals (we recall that we are dealing with a planar section of a larger three-dimensional swarm). In particular, 480 of them are uninformed followers, while the remaining ones are scouts with an initial streaker role, i.e. s i (0) = S for i ∈ {481, . . . , 500}. We here recall that uninformed bees are not allowed to change state, whereas the informed insects can switch between the streaker and the passive leader status. The swarm is initially arranged in an almost round area centred at (100 m, 100 m) of radius equal to r 0 = 4 m, where the positions of the insects are randomly assigned, refer again to Figure 4. In this respect, we account for a reasonable density of ≈ 8 bees/m 2 [35]; also the percentage of informed individuals, i.e. 4 % of the entire population, is in agreement with the experimental literature [36][37][38]. The initial gazing direction g i (0) of each generic ith bee is randomly generated as well.
The objective of the swarm is to reach the target destination. In this respect, the numerical realizations are stopped as soon as the last insect touches a point of ∂ nest , i.e. at a time denoted with t F . The dynamics of the bee population resulting from different combinations of the individual behavioural hypotheses outlined in Section 2 are classified according to the following criteria: In particular, we account for N = 500 bees, with 480 follower individuals and 20 scouts. All the informed bees are initially assigned a streaker status. For representative purposes, the virtual insects will be hereafter indicated by rigid disks centred at their actual position. More specifically, we will use yellow circles for uninformed individuals, red circles for streaker scouts and blue circles for passive leader scouts. Definition 3.1: The swarm undergoes a directionally productive motion towards the nest if where x swarm (t) = ( N i=1 x i (t))/N is the centre of mass of the bee cloud and l nest is the distance function introduced in Equation (3).
The swarm undergoes a coherent and collective flight if where a swarm (t) is a measure of the space extension 1 of the bee cloud at time t and a R is equal to π(r 0 ) 2 , i.e. the area of the round region initially containing all individuals. The swarm undergoes a collective and productive flight towards the nest if both conditions (25)-(26) are satisfied.
The quantity defined in Equation (26) is able to give an indication of the presence of dispersed insects, i.e. of bees unable to correctly synchronize their movement with the rest of the group. In fact, our simulation results have consistently shown that if, at a given time t, the surface of the insect cloud is twice the reference initial area a R (or even more), then there is at least a follower bee that actually has no groupmate within its alignment set which means that it has lost. Figures 5-7 summarize the results obtained by the different combinations of the individual behavioural hypotheses. In particular, for each pair of assumptions, we have run 10 independent numerical realizations, given the presence of randomness both in the initial position of the bees and in their dynamics due to the velocity term v rand . We can observe that in all cases the distance l nest (x swarm (t)) between the swarm centre of mass and the nest decreases almost linearly (see also the inset graph in Figure 5(a)).
In particular, it becomes null in a finite time (< 14 s), as E swarm is zero. However, as shown in the plot of C swarm in Figure 5(b), only under hypotheses (A3, L1), (A3, L2) and (A4, L2), the flight of the insect population is completely synchronized and therefore collective in all realizations. In the other cases, as also shown in the representative snapshots in Figures 6 and 7, at least one uninformed bee does not correctly align to the rest of the swarm, flying away from the groupmates. In these situations, the scout bees have to take time to reach/wait for the dispersed individual(s) and to guide it (them) towards the nest: the experimentally-observed compactness of the insect cloud is therefore not maintained. From the graph in Figure 5(b), it can be further noticed that bee dispersion is more significant in the case of assumptions (A2, L1) and (A2, L2), i.e. when the uninformed bees align to the faster groupmates, regardless of their status.
Taking all the results together, we can state that in our model alignment mechanisms involving a control over groupmate velocity do not certainly imply directionally productive and collective swarm dynamics. An efficient and coordinate flight is instead reproduced if the follower individuals synchronize their movement (i) to all insects sufficiently close to their position, regardless of their status and of the behaviour of the passive leaders, or (ii) only to close enough uninformed and streaker groupmates, provided the fact that the passive leaders stop upon reaching the front of the cloud. In particular, it is somehow  Representative evolutions of the bee population in the case of combinations between the hypothesis L1 (relative to the behaviour of the passive leader scouts) and the alternative assumptions on the alignment mechanism of the follower insects. It is possible to notice that, in all cases, the swarm reaches the target destination, but only for the pair (A3, L1) without bee dispersion. We recall that yellow disks represent follower bees, red circles represent streakers, and blue disks represent passive leaders. For each scout individual, we finally indicate by a coloured arrow its velocity.

Figure 7.
Representative evolutions of the bee population in the case of combinations between the hypothesis L2 (relative to the behaviour of the passive leader scouts) and the alternative assumptions on the alignment mechanism of the follower insects. It is possible to notice that, in all cases, the swarm reaches the target destination, bee dispersion is not observed for the pairs (A3, L2) and (A4, L2). We recall that yellow disks represent follower bees, red circles represent streakers, and blue disks represent passive leaders. For each scout individual, we finally indicate by a coloured arrow its velocity.
worth to remark that, in the case of hypotheses A1 and A4, passive leader bees are not considered for the alignment mechanism: however, they determine the flight of the follower groupmates by affecting the other velocity components (e.g. collision avoidance and attraction).
We now turn to describe and compare in more details selected characteristics of swarm dynamics under the plausible combinations of behavioural assumptions. In this respect, Figure 8 shows the time-evolution of the amount of bees belonging to each subpopulation in the different cases.As it is possible to observe, the number of follower individuals remains obviously constant whereas, transitions between streaker and passive leader states continuously occur. In particular, under the coupled hypotheses (A3, L1), the amount of streaker insects remains substantially higher that the number of passive leaders during the entire migration (the former falling within the range 14-20, the latter within the range0-6). On the opposite, in the case of assumptions (A3, L2) and (A4, L2) (i.e. when the passive  leaders are assumed to stop and wait for the rest of the colony), the fluctuations in the cardinality of two subgroups of scout bees are more significant: for instance, the amount of streakers can drop and be almost equal to the number of passive leaders. Figure 9 finally compares the trajectories of representative scout bees under the different assumptions resulting in realistic swarm dynamics. In particular, under the coupled hypotheses (A3, L1), it is straightforward to notice the short-time backward movement of the informed individual during its passive leader status. It is also interesting to observe that under the assumptions (A3, L2) and (A4, L2), the scout bees, when passive leaders, do not completely stop but rather still move as a consequence of the velocity components which are still active (i.e. those relative to attractive/repulsive interactions and to random fluctuations).
Variations in the percentage of scout bees. As seen, the collective migration of bee swarms is guided by few informed individuals, that are able to diffuse information of the nest location within the rest of the population. An interesting question is indeed relative to the consistency of the flight directional efficiency upon variations in the ratio between the number of individuals having the different roles within the colony.  (25)) and the percentage of informed bees, in the case of different sizes of the colony (i.e. of overall number of components N). To avoid redundancy, we plot the outcomes obtained from a single realization in the case of coupled hypotheses (A4, L2): however, these results have been obtained also with assumptions (A3, L1) and (A4, L1) and are robust in the case of independent simulations.
In this respect, we now study the dynamics of insect clouds characterized by different numbers of scouts and of overall components. In particular, we employ the same domain of the previous section and the rules of motion defined in Section 2: however, only the combinations of assumptions that have resulted in a plausible system phenomenology are hereafter used, i.e. the pairs (A3, L1), (A3, L2), and (A4, L2). As initial data, the bees are again randomly disposed in round regions, whose radius is chosen, in each case, to maintain the density of 8 bees/m 2 . All scout individuals have a streaker status whereas the initial bee gazing direction is randomly assigned.
As it is possible to observe in Figure 10, for a given group size, i.e. for a fixed N, the directionally productive component of swarm movement increases (i.e. E swarm decreases) as the percentage of informed individuals increases. Furthermore, still from the same plot, we can notice that the higher the overall number of bees is, the smaller the proportion of informed individuals necessary is in order to have an efficient migration towards the nest. In particular, substantially large swarms require a very small set of scout bees to reach the target destination.
These results are observed for all the tested combinations of behavioural assumptions (even if for clarity we represented only the case relative to the pair (A4, L2), being completely robust also in the case of independent simulations in the different settings (in each case, the standard deviation deriving from 10 simulations is < 1% and therefore not represented in the graphs). We also remark that in the cases of full productive swarming (i.e. when E swarm ≈ 0) no bee dispersion occurs (i.e. C swarm < 2), in agreement with the outcomes presented in the previous subsection.
As it will be commented in more details in the conclusive part of the paper, analogous quite surprising results have been obtained in the works by Couzin et al. [14] and by Fetecau and Guo [22], where similar microscopic/discrete models have been employed to describe collective swarming. Figure 11. Bee swarming in the case of more complex environments. Representative evolutions of the bee population placed within a domain characterized by a square obstacle between its initial position and the new nest, whose centre is located at the same y-coordinate of the initial centre of mass of the insect cloud, in the case of the sets of plausible behavioural assumptions (A3, L1), (A3, L2), and (A4, L2). The insect population is assumed to be composed of N = 500 individuals, which are initially subdivided into 480 uninformed insects and 20 streakers. The initial configuration of the swarm consists of a circle of radius r 0 = 4 m with bee position and gazing direction randomly assigned. It is possible to observe that the swarm autonomously deflects its motion and undergoes morphological reorganization in order to pass the structural element and compactly reach the target destination. We recall that yellow disks represent follower bees, red circles represent streakers, and blue disks represent passive leaders. For each scout individual, we finally indicate by a coloured arrow its velocity.

Swarming in more realistic situations
Finally, we turn to asses the applicability of our model in real-world situations by means of representative numerical results involving more complex environments.
In particular, referring to Figures 11 and 12, we deal with domains characterized either by a structural obstacle or by a bottleneck, placed in between the initial position of the swarm and the target destination. Such environmental elements may represent architectural buildings or trees that the insect cloud has to avoid during its migration. Hereafter, the bee population is still assumed to be composed of N = 500 individuals, which are initially subdivided into 480 uninformed insects and 20 streakers. As usual, the initial configuration Figure 12. Bee swarming in the case of more complex environments. Representative evolutions of the bee population placed within a domain with a bottleneck between its initial position and the new nest, whose centre is located at the same y-coordinate of the initial centre of mass of the insect cloud, in the case of the sets of plausible behavioural assumptions (A3, L1), (A3, L2), and (A4, L2). The insect population is assumed to be composed of N = 500 individuals, which are initially subdivided into 480 uninformed insects and 20 streakers. The initial configuration of the swarm consists of a circle of radius r 0 = 4 m with bee position and gazing direction randomly assigned. It is possible to observe that the swarm squeezes to pass through the structural element. We recall that yellow disks represent follower bees, red circles represent streakers, and blue disks represent passive leaders. For each scout individual, we finally indicate by a coloured arrow its velocity.
of the swarm consists of a circle of radius r 0 = 4 m with bee position and gazing direction randomly assigned. Again we test the coupled hypothesis (A3, L1), (A3, L2), and (A4, L2). The centre of the nest is located at the same y-coordinate of the initial centre of mass of the insect cloud.
As it is possible to observe in Figures 11 and 12, in both situations, the swarm has to slightly deflect its direction of movement and to deform to pass the structural elements and reach the target destination. In particular, in the case of the square obstacle, the bees located at the bottom part of the population are pressed towards the centre of the swarm by the repulsive velocity v boundary . We indeed have an increasing density of insects in the centre of the cloud. However, the productive direction of flight is still maintained. Once passed the structural element, the compressed area of the swarm slightly relaxes and an almost homogeneous density of bees is recovered.
Referring to Figure 12, when approaching the bottleneck, the swarm is instead substantially stretched horizontally and compressed vertically, i.e. it switches from a round shape to an ellipsoidal geometry, with shorter axis along the y-direction of the domain. Interestingly, after passing the bottleneck, there is only a slight relaxation of the insect cloud, which does not acquire again a fully round configuration. The underlying rationale is that, even when the colony has an elongated shape, the component bees are at a sufficient (but not excessive) distance one from another and therefore there is no reason to spend energy to further reorganize.
In both domains configurations, the swarm finally redirect again its coordinate flight to reach the target destination.
The above-described phenomenologies are observed under the three tested coupled hypotheses, with slight differences in the morphological transitions of the swarm. However, we remark that, in all settings, there is no bee dispersion.
The results presented in this section allow us to conclude that the behavioural rules of bees (A3, L1), (A3, L2), and (A4, L2) give a realistic swarming not only in open-space simple domains but also in more complex scenarios. In particular, our model, under such plausible hypotheses, is able to capture autonomous morphological reorganizations and changes of flight direction of the insect cloud, necessary to preserve its compactness and to reach the target destination.

Discussion
The collective motion of groups of animals has recently attracted an increasing interest in the modelling community. Particularly intriguing is the swarming of bee populations towards a new nest. In fact, such a characteristic migration is led by few informed/scout individuals, which have previously explored the possible new home and therefore are able to guide the groupmates towards the target destination. However, also from an experimental point of view, little is known on the behavioural rules underlying the coordinated flight of bee swarms. In particular, it is not completely clear how the information on the nest location spreads within the population and what are the exact dynamics of the informed individuals.
The aim of this work has been indeed to test alternative assumptions and to find out those resulting in realistic swarming phenomenologies. To do this, we employed a discrete mathematical model, where each insect has been individually represented by a material point and assigned a given behavioural status (i.e. streaker, passive leader, or uninformed). The subgroups of bees have in common some general rules of motility, such as the tendency to remain within the population while keeping a comfort distance from the other components. Other principles of motion are instead characteristic of a single subpopulation, such as the ability if uninformed bees to synchronize their flight with the surrounding groupmates. These ideas have been translated in a mathematical model based on a set of first-order ODEs, each of them describing the evolution of the position of an insect.
The resulting model has been used to test combinations of alternative assumptions underlying the synchronization mechanisms of uninformed bees and the individual behaviour of passive leader bees. In particular, our results have shown that a productive collective flight of the swarm is only possible if the uniformed individuals synchronize their movement (i) to all insects sufficiently close to their position regardless of their status and velocity and of the dynamics of the passive leaders (i.e. coupled assumptions (A3, L1) and (A3, L2)) or (ii) only to close enough follower and streaker groupmates, provided the fact that the passive leaders stop upon reaching the front of the swarm (i.e. coupled assumptions (A4, L2)). Other sets of hypotheses have produced the unrealistic phenomenon of bee dispersion, i.e. at least one follower individual is not able to synchronize its movement with the rest of the swarm during the entire flight, thereby flying away and affecting the migration of the informed bees and eventually of the entire population.
Once the most plausible behavioural assumptions have been identified, we have turned to analyse the effect of variations in the number of scout bees. Interestingly, we have found that larger swarms require fewer scout individuals to compactly reach the target destination. This quite surprising outcome is in agreement with the results obtained by similar models [14,22]. However, it is useful to remark that from experimental observations it is known that, regardless the size of the population, the fraction of informed insect typically falls in the range 3-5%In this respect, we can speculate that, although in principle the percentage of scout bees could decrease, their amount may be established also by other social dynamics of the swarm not involving migration issues. For instance, a sufficiently high number of scout bees could be necessary to explore the environment to find a new home in a substantially short time.
With the last set of simulations, we have finally provided the fact that our model, with the selected combinations of bee behavioural assumptions, is able to capture swarm dynamics in more complex scenarios, that may require morphological rearrangements of the insect cloud to pass structural elements and significant changes of flight directions.

Comparison with pertinent literature
The description of the collective and coordinated dynamics of groups of animals is a challenging topic for theoretical researchers. Populations of intelligent living entities are in fact complex systems, since the component individuals are not passively dragged by external forces but rather they undergo active decision-based dynamics, so that the use of classical passive mechanics is no longer sufficient. The overall evolution of the group then emerges from the rules governing the individual behaviour. In this respect, the mathematical and computational literature in this field presents indeed a wide range of approaches.
For instance, microscopic models (also called individual-based models, IBMs) describe a group of animals as a collection of isolated agents: each of them is individually considered, assimilated for instance to a point particle or a quasi-rigid disk and followed during motion. More specifically, a first subgroup of microscopic models is represented by the so-called cellular automata (CA), where each animal is set to behave according to phenomenological algorithmic rules, that depend on its individuality and/or on the surrounding environment. Another subtype of microscopic approach involves instead discrete models: they rely on classical Newtonian laws of point mechanics, as the motion of each agent is defined by a first-or a second-order ordinary differential equation (ODE).
However, when the number of component individuals is significantly large, as in the case of fishes [30] or myxobacteria [24,26], microscopic methods become computationally expensive and therefore different approaches are needed. In this respect, continuous models, characteristic of a macroscopic point of view, rely on the definition of a proper density of agents, which evolves following (typically nonlinear) partial differential equations (PDEs), which implement conservation laws and require phenomenological assumptions for their closure, see for example [3,[41][42][43].
A bridge between the microscopic word and the macroscopic representation of animal systems is represented by kinetic models. Characteristics of a mesoscopic point of view, they are able to derive, employing hydrodynamic arguments, Boltzmann-like evolution laws for statistical distribution functions, which describe position and velocity of the components of the population of interest [5,7,27].
The model presented in this article belongs to the class of microscopic/discrete methods. Since some of them are devoted to reproduce selected features of bee swarming, it is important to discuss differences and similarities with respect to our approach, mainly in the term relative to the behaviour of scout/informed individuals. Entering in more details, in [22], Fetecau and Guo implement a second-order model with differentiation between informed and uninformed bees. More specifically, the formers do not interact with their groupmates and move faster towards the target destination according to two alternative hypothesis: (i) they streak with a constant acceleration or (ii) they fly with a constant speed. In both cases, when such informed individuals reach the leading edge of the cloud, they come back to the rear of the population being 10 time less visible. Each uninformed insect instead undergoes attractive and repulsive stimuli, described by a Morse potential, and alignment mechanisms, which involve its two-fold faster neighbours. Bee dynamics account for a random component as well, which is active only when the interaction of an individual with the rest of the swam is low enough. The authors also introduce a visual field for each bee, given by a planar cone which is constantly aligned to the direction of motion and formed by two regions: a central cone where the other individuals are set to be seen directly, and therefore assigned a unit weight, and a peripheral area where the other individuals are set to be seen partially, and therefore assigned a lower weight. In our model, a peripheral vision is not considered, since it is known from biology that the compound eyes of bees cover most of the front and of the sides of their head, assuring an almost homogeneous vision. The model by Fetecau and Guo is therefore based on behavioural rules similar to the pairs (A1, L1) and (A2, L1) employed in our article. Interestingly, in both works, such hypotheses result in a directionally efficient swarming: however, we have discarded the two combinations of assumptions as a consequence of the lack of a consistent compactness of the insect cloud.
As in the case of the work by Fetecau and Guo, also in [28], a set of informed bees is defined within the swarm and assigned a back-and-forward motion within the swarm, in order to diffuse the information of the productive direction to the overall population. Such a first-order model also involves attraction, repulsion, and alignment. In particular, the cohesion velocity contribution is modelled as a vector pointing from the position of each bee to the centre of mass of the set of neighbouring insects which fall within its visual distance. In this respect, we here preferred to implement pairwise interaction kernels, since it is difficult to establish whether a bee exactly knows the position of the centre of mass of the rest of groupmates. The alignment rule instead relies on an Euclidean metricbased assumption, namely each bee is set to synchronize its movement with all the seen groupmates (regardless of their speed).
In [20], the authors describe both the decision-making process used by the househunting honeybees to find a new nest site and their guidance role within the rest of the swarm. Focusing on the latter, we can notice that Diwold and colleagues employ a cohesion term that makes each bee attracted by the centre of mass of the population only in the case of the presence of at least a fast enough individual within a given neighbourhood. These authors also implement a topological metric-based alignment mechanism, i.e. a flight synchronization with a given number of closest fast individuals. The resulting model is then applied to compare the swarming of two different species of honeybees, namely Apis Mellifera and Apis Florea. In particular, while A. Mellifera is a cavity-nesting species, whereas A. Florea is an open-nesting species. This means that the Apis Mellifera has to find a roomy and comfortable homesite, protected from cold winds and from predators. Conversely, A. Florea usually nests on a shaded branch, having less constraints in finding a suitable location.
A more general (i.e. not strictly related to bee dynamics) model is proposed in [14]. The authors here focus on two aspects: how information is transferred among moving groups of animals and how they can find an agreement when scout individuals suggest different moving directions to rest of the population. Such an approach still relies on the classical social principle of attraction/repulsion and alignment. As previously commented, one of their main results is related to the relationships between the percentage of informed bees and the flight efficacy of the swarm at different population sizes. The same research group also proposes a model that focuses on the pattern characteristic of animal populations [15]. Their approach includes a Morse potential and two additional terms: they are relative to self-propulsion and friction and their balance results in the capability of the system to reach an asymptotic collective speed (as it happens also in our model in given regimes of the free parameters).
Finally, the Cucker-Smale model (even with the inclusion of temporary leader emergence) accounts only for a term relative to a movement synchronization mechanism, which is affected by a communication rate that depends on the interindividual mutual distances [17,18]. The model is, of course, able to capture flocking phenomenology.
Chen and Kolokolnikov instead presented in [10] a particle model able to capture selected features of predator-prey interactions in the case of a generic swarm. In their approach, each prey is attracted by its groupmates, whereas repulsive stimuli describe its intention to maintain a comfort space within the insect cloud and to fly away from the predator. The predator, in turn, is attracted by the preys. In more details, hyperbolic laws are implemented to describe repulsive dynamics (as in our model) while a linear short-range attraction is used for the individual cohesive behaviour. The approach by Chen and Kolokolnikov is finally able to predict the shape of the swarm as well as the behaviour of the predator in different regions of the model parameter space.
By reviewing the different works commented in this dissertation, we can conclude that the main features of a swarming phenomenology can be captured by minimal, i.e. two-or three-component, models. In fact, reasonable configurations of swarms (where the agents stabilize at given and finite mutual distances) can be obtained only by taking into account repulsive/attractive pairwise dynamics. The insertion of an aligned mechanism, such as the one we proposed in [2], is instead needed to get effective directional flights. Such simple models have also the advantage of being suitable for interesting analytical analysis and insight, e.g. on the properties of the steady states of the system as done by different groups [7,8,23].
In this respect, the inclusion of more sophisticated model ingredients, i.e. bee status differentiations and relative transitions and flight rules, is therefore not essential to reproduce basic collective dynamics of insect swarms. However, such model components have been here introduced in order to be as close as possible to experimental evidences and to find out reasonable assumptions at the basis of the still unknown bee behaviour.

Further developments
The proposed model has investigated and tested different social mechanisms underlying the behaviour of a bee swarm. However, our approach can be further improved in several directions. First, it would be useful to have a better comparison with experimental data. In particular, this would improve the quality of the work from two points of view: it could be possible to derive a more precise parameter estimate and we could have a quantitative validation of the proposed theoretical results.
Further, a three-dimensional extension of the model would be a natural development. In fact, it would allow to better describe the dynamics of the informed bees which, when passive leaders, are also supposed to hide themselves along the top or the bottom region of the swarm in order to not significantly affect the flight of the groupmates, as hypothesized in the empirical literature [33,35] and previously commented in this work.

The spatial extension of the swarm is evaluated by the Matlab (the MathWorks ) function
boundary. More specifically, this function returns the area enclosed by the single conforming 2-D boundary containing of a given set of discrete particles. For further details, we refer to Matlab manuals and tutorials. 2. From statistical mechanics [32], a system of mutual interacting particles is said H-stable if, for any arbitrarily large number of agents, the microscopic agents will not collapse onto themselves and a typical distance between individuals will be well defined.

Appendix. Parameter estimate
The proposed approach is intrinsically multiparametric. In particular, the model coefficients can be classified in two groups: those that have a direct and measurable biological meaning (e.g. speed values) and those that are more technical, i.e. that only subsume experimental dynamics, such as the interaction coefficients. We indeed derived a composite parameter setting, obtained by observations and data present both in the experimental and in the theoretical literature and, when necessary, by preliminary numerical realizations. First, the half visual angle θ vis , which symmetrically extends from the individual gazing direction, was taken equal to 156.5 • , according to the biological measures presented in [40]. We here remark that a visual field determined by such an angle θ vis (i.e. < 180 • ) introduces anisotropy in the behaviour of bees although, with respect to most animal species, they are characterized by a substantially limited blind area behind them. As far as we know, in the experimental literature there was instead no study that explicitly defined the depth of the bee visual field. We therefore opted to set d vis = 20 m, which is a value that allows each insect to perceive the presence of all groupmates when the swarm is sufficiently compact and not dispersed. The proposed estimate took also into account the domain characteristic dimensions, such as the distance of the nest from the initial position of the insect population: in this respect, d vis was set small enough to avoid that the target destination falls within the visual field of the uniformed individuals at the beginning of the observation time.
In our model two characteristic bee speed values are taken into account. The maximal admissible velocity v max , introduced in Equation (7) was set equal to 9.4 m/s in accordance with [35]. The mean speed of the uninformed insects, defined in Equation (16), was instead fixed equal to 6.7 m/s, again coherently with the experimental literature [25]. We here underline that v max is also used in the case of the fast fly of the streaker bees towards the nest, whereas v mean for the slower backward movement of the passive leader towards the trailing edge of the swarm. The difference between the speed of streakers and of the uninformed individuals has been empirically demonstrated and is at the basis of the rejection of the subtle flight alignment hypothesis [35].
All bees, regardless of their status, are characterized by repulsive/attractive interactions. According to the measure reported in [35], the insects tend to preserve a minimal mutual distance, here denoted by d avoid , equal to 0.3 m. The extension of the alignment region d align was taken equal to 2 m. Since this value can not be empirically measured, we obtained its estimate referring to the modelling literature. The ratio d avoid /d align used in this work falls in fact in the middle of the range of analogous quantities tested by Couzin and colleagues [15]. A d group = d vis was instead set since we have assumed that each insect aims to maintain a connection with all individuals within its visual region.
As already commented in the text, from a mathematical point of view, f avoid (with units m 2 /s) gives the slope of the hyperbolic repulsive kernel h avoid , whereas f group (with units m/s) establishes the maximum of the parabolic-type attraction behaviour of bees, described by function h group . Both positive parameters indeed do not have a clear and direct experimental counterpart and therefore their estimate was not straightforward. However, a numerical study, supported by selected empirical evidences, facilitated in this respect. In particular, we first took into account the following considerations: (i) upon attrative/repulsive stimuli only (in the absence of directional, alignment, and random dynamics), the computational swarm has to stabilize in a realistic crystalline configuration, characterized by optimal interparticle spacing ≈ d avoid (i.e. we have to avoid unrealistical situations such as the collapse or the explosion of the insect cloud); (ii) the specific flight of the informed bees has not to be affected by other velocity contributions, in the case of both assumptions L1 and L2.
To account for observation (i), we ran a series of numerical realizations varying the interaction parameters f avoid and f group in the case of a swarm formed by N = 500 bees (480 of them with a follower role and the remaining 20 with an initial streaker status), which were assumed to be subject only to repulsive/attractive stimuli, i.e.
for all i = 1, . . . , N, where the interaction velocity components were defined as in Equations (11)- (12). Given the same domain and initial conditions described in Section 3.1, the obtained dynamics were then classified according to the following asymptotic quantities: begin t f an observation time sufficiently large to allow the insect cloud to reach a stable equilibrium configuration. The measures introduced in (A2)-(A3) have clear empirical meanings: d min is in fact the minimal interparticle distance, whereas d max defines the extension of the overall swarm. As shown by the representative cases reported in Figure A1, almost all pairs of coefficients (f avoid , f group ) such that f avoid /f group ≥ 10 6 resulted in realistic swarm pattern, as d min was very close to the experimentally measured bee comfort space d avoid . In these cases, also the swarm overall diameter d max was consistent with the empirical observations relative to the spatial density of bees [35]. On the opposite, if f avoid /f group < 10 6 the insects were observed to stabilize unrealistically close one to another, as d min d avoid . Such simulation outcomes were indeed able to give a first restriction of the possible variations of the interaction coefficients f avoid and f group . Interestingly, the resulting permitted interaction parameters fall within the H-stability region 2 of the space of interaction parameters (f avoid , f group ) that would characterize pairs of attractive/repulsive interaction kernels analogous to h avoid and h group , with the same coefficients d avoid and d group , if we neglected the asymmetry introduced by the anisotropic visual region of bees (cf. Hypothesis 3 in [4] is not satisfied in our case).
In fact, referring to the series of works by Carrillo and colleagues [4][5][6][7][8][9] (in particular, [9] deals with particle-based models of swarming), and to the calculations proposed in Section 3.2 in [2], we have that the H-stability region for the swarm of our interest in the case of fully isotropic hypotheses (i.e. if the bees had a round visual field) would be given by the following parametric relation: 2 3(d avoid ) 2 + 4d group d avoid + 3(d group ) 2 = 1.0719164 · 10 5 , (A4) which is indicated by the grey-shadowed area in Figure A1 that contains the couples of permitted parameters found by the above numerical investigation. However, despite the consistence between these analytical results and the obtained computational outcomes, it is useful to underline that a theoretical analysis of the H-stability properties of an agent-based system in the case of asymmetric attractive/reulsive kernels is far to be provided and therefore would require further investigations.
Having reduced the range of values of the interaction parameters, we then used the above-cited observation (ii) to have a further estimate. In this respect, we varied the coefficients f avoid and f group in the case of a simulation setting involving a swarm formed again by 480 follower individuals and 20 streaker bees. The insect population, placed in the open-space domain with initial conditions defined again as in Section 3.1, was assumed to behave according to the following rules: for i = 1, . . . , 500, being the velocity contributions defined exactly as in Section 2 (in particular v passive i was set to take the form either of Equation (23) or of Equation (24)). With respect to the complete model, we indeed neglected alignment mechanisms and random contributions. Our choice was justified by the fact that the aim of the study was to find the values of the attraction/repulsion parameters that did not affect the characteristic motion of the informed bees, under the assumptions L1 and L2. As summarized in Figures A2 and A3, we observed that, in both cases, too large values of f group disrupted the hypothesized flight of the informed bees, regardless of the value given to f avoid . In more details, the group of following bees constantly stabilized into a crystalline configuration but, for f group > 10 −3 , the following dynamics arose: • in the case of assumption L1, the scout individuals were not able to reach any edge of the fixed cloud (see the bottom-right panel of Figure A2); • in the case of assumption L2, the informed insects were not able to rest at the leading front of the population (where they had to remain since the follower bees had not allowed to have a directional movement and therefore to pass the scouts), as reproduced in the bottom-right panel of Figure A3.
The underlying rationale involves two competing mechanisms: on one hand, a too large attraction strength f group makes the group of follower individuals almost a rigid disk which is difficult to be flown across; on the other hand, it causes the scout bees to perform an abnormal movement.
Within the remaining set of permitted interaction parameter values, we finally opted to fix f avoid = 1 m 2 /s, i.e. we opted for a classical equilateral hyperbolic repulsive kernel as done, for instance, by Kolokolnikov and Chen in the already cited work [10] dealing with predator-prey swarming dynamics. An f group = 10 −6 m/s was consequently set to avoid further increments in the difference between the order of magnitude of the two parameters.
As seen, a repulsive velocity component from the domain boundary, given by a negative exponential function, has been set for each bee. In this respect, the insects perceive and react to the presence of the boundaries from a distance of d boundary , whereas a boundary and b boundary determine the exact form of v boundary i . In the absence of pertinent experimental measurements, a reasonable estimate of such triplet of coefficients was obtained with a series of preliminary simulations, i.e. in order to avoid Figure A1. Dependence of the stable configuration of the bee cloud, subjected only to attractive and repulsive dynamics, upon variations in the values of the interaction parameters f avoid and f group . The quantities d min and d max represent the minimal interagent distance and the overall diameter of the swarm at a observation time t f sufficiently large to have a stabilization of the system, as defined in Equations (A2) and (A3), respectively. The grey area in each panel indicates the H-stability region that one would have in the case of the same interaction kernels by assuming a isotropic visual region of bees. Figure A2. Dependence of the dynamics of the informed bees upon variations in the values of the interaction parameters f avoid and f group , in the case of assumption L1. As it is possible to observe the hypothesized 'back-and-forth' flight can be obtained only for f group < 10 −3 , regardless of the values of coefficient f avoid (provided that the pair (f avoid , f group ) leads to crystalline equilibrium configurations upon attractive/repulsive interactions only). Too large values of f group in fact result in a disrupted behaviour of the informed insects, which remain stuck within the bee cloud (represented by the yellow shadow), as reproduced in the corresponding representative bottom panel. Figure A3. Dependence of the dynamics of the informed bees upon variations in the values of the interaction parameters f avoid and f group , in the case of assumption L2. As it is possible to observe the hypothesized 'go-and-stop' flight can be obtained only for f group < 10 −3 , regardless of the values of coefficient f avoid (provided that the pair (f avoid , f group ) leads to crystalline equilibrium configurations upon attractive/repulsive interactions only). Too large values of f group in fact result in a disrupted behaviour of the informed insects, which are not able to stop at the leading edge of the bee cloud (represented by the yellow shadow) being dragged within the population, as reproduced in the corresponding representative bottom panel.
unrealistic dynamics such as swarm collapse at the domain boundary or deflection from the optimal flight trajectory at too large distances from the domain edge. In particular, the found values of a boundary and b boundary are analogous to their counterpart employed in a particle model reproducing pedestrian behaviour [13]. For the sake of completeness, we illustrate in Figure A4  Finally, the modulus of the random velocity vectors falls within the range [0, v mean /10]: in particular, we set such an upper threshold to avoid unrealistically large fluctuations in bee flight, taking also advantage of the study and relative observations performed in [2].